The projective Lorentzian space is a closed Euclidean ball

Let $\mathrm{H}^d_n$ be the space of degree-$d$ homogeneous polynomials in $n$ variables, and let $\mathrm{L}^d_n$ be the cone of Lorentzian polynomials in $\mathrm{H}^d_n$. Let $\mathbb{P}\mathrm{L}^d_n$ denote the image of $\mathrm{L}^d_n$ in the projectivization of $\mathrm{H}^d_n$. Ball conjecture. The space $\mathbb{P}\mathrm{L}^d_n$ is homeomorphic to a closed Euclidean ball. The space is already known to be compact and contractible; the conjecture identifies its full topological type.